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Ann. Sci. Ecole Norm. Sup. (4)

(to appear)

ANDRE-QUILLEN HOMOLOGY OF ALGEBRA RETRACTS

LUCHEZAR L. AVRAMOV AND SRIKANTH IYENGAR

Abstract. Given a homomorphism of commutative noetherian rings ϕ : R → S,Daniel Quillen conjectured in 1970 that if the Andre-Quillen homology functorsDn(S |R;−) vanish for all n � 0, then they vanish for all n ≥ 3. We prove theconjecture under the additional hypothesis that there exists a homomorphism ofrings ψ : S → R such that ϕ ◦ψ = idS . More precisely, in this case we show that ψis complete intersection at ϕ−1(n) for every prime ideal n of S. Using these results,

we describe all algebra retracts S → R → S for which the algebra TorR• (S, S) is

finitely generated over TorR0 (S, S) = S.

HOMOLOGIE D’ANDRE-QUILLEN DES ALGEBRES SCINDEES

Resume. Etant donne un homomorphisme ϕ : R → S d’anneaux commutatifsnoetheriens, Daniel Quillen a conjecture en 1970 que si les foncteurs Dn(S |R;−)d’homologie d’Andre-Quillen sont nuls pour tout n � 0, alors ils sont nuls pourtout n ≥ 3. Nous demontrons cette conjecture sous l’hypothese supplementairequ’il existe un homomorphisme d’anneaux ψ : S → R tel que ϕ ◦ ψ = idS . Plusprecisemment, nous montrons que dans ce cas ψ est d’intersection complete enϕ−1(n) pour tout ideal premier n de S. En utilisant ces resultats, nous decrivons

toutes les algebres scindees S → R → S pour lesquelles l’algebre TorR•(S, S) est

finiment engendree sur TorR0 (S, S) = S.

Date: October 14, 2001; revised April 14, 2002.1991 Mathematics Subject Classification. Primary 13D03, 14B25. Secondary 13H10, 14M10.L.L.A. was partly supported by a grant from the N.S.F.S.I. was supported by a grant from the E.P.S.R.C.

1

2 L. L. AVRAMOV AND S. IYENGAR

Introduction

Let ϕ : R→ S be a homomorphism of commutative noetherian rings.For each n ≥ 0, let Dn(S |R;−) denote the nth cotangent homology functor on

the category of S-modules, defined by Andre [1] and Quillen [25]. To study howvanishing of these Andre-Quillen homology functors relates to the structure of ϕ,we define the Andre-Quillen dimension of S over R to be the number

AQ-dimR S = sup{n ∈ N | Dn(S |R;−) 6= 0} ;

in particular, AQ-dimR S = −∞ if and only if Dn(S |R;−) = 0 for all n ∈ Z.Vanishing of Andre-Quillen homology in low dimensions characterizes important

classes of homomorphisms of noetherian rings. Recall that ϕ is regular if it is flatwith geometrically regular fibers. It is etale if, in addition, it is of finite type andunramified. A general locally complete intersection, or l.c.i., property is defined in7.2; when ϕ is of finite type, it means that in some (equivalently, every) factorizationof ϕ as an inclusion into a polynomial ring followed by a surjection, the kernel ofthe second map is locally generated by a regular sequence. The following resultswere proved in [1], [25] for maps ϕ of finite type, and in [4], [10] in general:

(A) AQ-dimR S = −∞ and ϕ is of finite type if and only if ϕ is etale.(B) AQ-dimR S ≤ 0 if and only if D1(S |R;−) = 0, if and only if ϕ is regular.(C) AQ-dimR S ≤ 1 if and only if D2(S |R;−) = 0, if and only ϕ is l.c.i.

Further research on homomorphisms of finite Andre-Quillen dimension has beendriven by two conjectures, stated by Quillen in 1970. One of them, [25, (5.7)], isfor maps locally of finite flat dimension: For each prime ideal n of S the R-moduleSn has a finite resolution by flat R-modules. That conjecture was proved in [10]:

(D) AQ-dimR S <∞ and ϕ is locally of finite flat dimension if and only if ϕ is l.c.i.

As a consequence, if ϕ is locally has finite flat dimension, then AQ-dimR S <∞implies AQ-dimR S ≤ 1. The remaining conjecture, [25, (5.6)], predicts the behaviorof Andre-Quillen dimension when no flatness hypothesis is available.

Quillen’s Conjecture. If AQ-dimR S <∞, then AQ-dimR S ≤ 2.

No structure theorem is known for R-algebras S with AQ-dimR S ≤ 2, so theconjecture presents a significant challenge beyond the generic difficulty of computingthe modules Dn(S |R;M), defined in terms of simplicial resolutions. This partlyexplains why so few cases have been settled. In [10] the conjecture is proved whenone of the rings R or S is locally complete intersection. Indirect evidence is obtainedin [21]: If ϕ is a large homomorphism of local rings in the sense of [23], R hascharacteristic 0, and AQ-dimR S is an odd integer, then AQ-dimR S = 1.

Our main result establishes Quillen’s Conjecture when S is an algebra retract ofR, meaning that there exists a homomorphism of rings ψ : S → R such that ϕ◦ψ =idS ; any homomorphism ψ with this property is called a section of ϕ. Algebraretracts frequently arise from geometric considerations. For instance, to study amorphism of schemes X → Y one often uses the induced diagonal embeddingX → X ×Y X . The underlying algebraic construction is the homomorphism ofrings ϕ : S ⊗A S → S defined by ϕ(s′ ⊗ s′′) = s′s′′; the ring S is an algebra retractof R = S ⊗A S, with section ψ(s) = s ⊗ 1. A different type of retracts arisesin constructions of projective schemes. They typically involve a graded S-algebraR =

⊕∞i=0 Ri with R0 = S; the relevant homomorphisms ϕ and ψ are, respectively,

the canonical surjection R→ (R/R>1) = S and the inclusion S = R0 ⊆ R.

HOMOLOGY OF ALGEBRA RETRACTS 3

An important aspect of our result is that it connects the homological conditionsin the conjecture through the structure of retracts of finite Andre-Quillen dimension.Let, as always, SpecS denote the set of prime ideals of S. If ϕ has a section ψ,then for every n ∈ SpecS one can find a set x of formal indeterminates over Sn andan ideal b contained in n(x) + (x)2 that fit into a commutative diagram

(En)

(Rm)∗

(ϕn)∗

%%KKKKKKKKKKK

Sn

ψ′

//

(ψm)∗99sssssssssss

Sn[[x]]/(b)

∼=

OO

ϕ′

// Sn

of homomorphisms of rings, where m = ϕ−1(n), asterisks ∗ denote (Ker(ϕ))-adiccompletion, ψ′ is the natural injection and ϕ′ the surjection with kernel (x).

For every real number c set bcc = sup{i ∈ Z | i ≤ c}.

Theorem I. Let ϕ : R→ S be a homomorphism of rings and set a = Ker(ϕ). If ϕadmits a section and R is noetherian, then the following conditions are equivalent.

(i) AQ-dimR S <∞ .(ii) AQ-dimR S ≤ 2 .(iii) D3(S |R;−) = 0 .(iv) Dn(S |R;−) = 0 for some n ≥ 3 such that bn−1

2 c! is invertible in S .(v) For each n ∈ SpecS, the ideal b in some (respectively, every) commutative

diagram (En) is generated by a regular sequence.

We apply the results discussed above in concrete cases, illustrating the knownfact that all dimensions allowed under Quillen’s Conjecture do occur.

Examples. Let x, y be indeterminates over S. The natural homomorphisms

S // S[x, y] // S[x, y]/(x2, xy, y2) // S[x]/(x2) // S

P R T

provide the following list of Andre-Quillen dimensions:

AQ-dimS S = AQ-dimP P = AQ-dimR R = AQ-dimT T = −∞

AQ-dimS P = 0

AQ-dimS T = AQ-dimP T = AQ-dimP S = 1

AQ-dimT S = 2

AQ-dimS R = AQ-dimP R = AQ-dimR T = AQ-dimR S =∞

Indeed, (A), (B), and (C) yield the equalities in the first three lines; (C) also impliesAQ-dimT S ≥ 2. Because S is a retract of T , Theorem I provides the converseinequality; since T and S are retracts of R, the theorem also computes the last twodimensions on the last line. The two remaining dimensions on that line are givenby (D), because R has finite flat dimension over S and over P .

We use Theorem I together with our results in [13] in a situation that doesnot a priori involve Andre-Quillen homology—the classical homology of an algebraretract S → R → S. In that case TorR0 (S, S) = S and TorR

•(S, S) is a graded-

commutative algebra with divided powers, but precise information on its structureis available in two instances only: when S is a field, cf. [22], [19], or when R→ S islocally complete intersection. Our second main result contains a description of allnoetherian algebra retracts with finitely generated homology algebra.


Let MaxS denote the set of maximal ideals of S.

Theorem II. Let Sψ−→ R

ϕ−→ S be an algebra retract with noetherian ring R, and

set Max′ S = {n ∈ MaxS | char(S/n) > 0}. The following conditions are equivalent.

(i) The S-algebra TorR•(S, S) is finitely generated.

(ii) For every S-algebra T there exists an isomorphism of graded T -algebras

TorR•(S, T ) ∼=

(∧SD1 ⊗S SymS D2

)⊗S T

where D1 and D2 are projective S-modules concentrated in degrees 1 and 2,respectively, and (D2)n

′ = 0 for all n′ ∈ Max′ S.(iii) The S-modules D1(S |R;S) and D2(S |R;S) are projective, D3(S |R;S) = 0,

and D2(S |R;S)n = 0 for all n ∈ Max′ S.(iv) For each n ∈ SpecS, the ideal b in some (respectively, every) commutative

diagram (En) is generated by a regular sequence contained in (x)2, andb = 0 if n is contained in some n′ ∈Max′ S.

If S is a flat algebra over some ring A, then TorS⊗AS• (S, S) is isomorphic to the

Hochschild homology algebra HH•(S|A) of S over A. Our main result in [13] showsthat if the ring R = S⊗AS is noetherian, and HH•(S|A) is finitely generated as analgebra over S, then S is regular over A. On the other hand, by the Hochschild-Kostant-Rosenberg Theorem [20], as generalized by Andre [3], if S is regular over Athen HH•(S|A) ∼=

∧SD1. Thus, in the context of Hochschild homology the module

D2 in Theorem II is trivial. It is also trivial for algebra retracts where all theresidue fields of S have positive characteristic. However, Q → Q[x]/(x2) → Q hasfinitely generated Tor algebra with D2 6= 0.

We proceed with an overview of the contents of the article. Although its maintopic is the simplicially defined Andre-Quillen homology theory, many argumentsare carried out in the context of DG (= differential graded) homological algebra.

Section 1 contains basic definitions and results on DG algebras.In Section 2 we recall the construction and first properties of non-negative in-

tegers εn(ϕ), attached in [10] to every local homomorphism ϕ. These deviations ,whose vanishing characterizes regularity and c.i. properties of ϕ, are linked to cer-tain Andre-Quillen homology modules, but are easier to compute. Section 3 con-tains a general theorem on morphisms of minimal models of local rings. Its proofis long and difficult. Its applications go beyond the present discussion.

The next two sections are at the heart of our investigation.In Section 4 we define a class of local homomorphisms, that we call almost

small . It contains the small homomorphisms introduced in [8], and its larger sizeoffers technical advantages that are essential to our study. We provide variouscharacterizations of almost small homomorphisms and give examples. The keyresult established in this section is a structure theorem for surjective almost smallhomomorphisms of complete rings in terms of morphisms of DG algebras.

The proof of Theorem I depends on another new concept—that of weak categoryof a local homomorphism. It is defined in Section 5, where arguments from [10] areadapted in order to obtain information on the positivity and growth of deviations ofhomomorphisms with finite weak category. To apply these results to almost smallhomomorphisms we prove that they have finite weak category; the proof involvesmost of the material developed up to that point.

In Section 6 we return to Andre-Quillen homology, focusing on local homomor-phisms of local rings. We show that vanishing of homology with coefficients in the


residue field characterizes complete intersection homomorphisms among the homo-morphisms having finite weak category. This leads to local versions of Theorems Iand II above. The theorems themselves are proved in the final Section 7.

The main results of this paper were announced in [14], cf. also Remark 7.6. Thatarticle provides historical background, a more leisurely discussion of applications ofAndre-Quillen homology to the structure of commutative algebras, and new proofsof some earlier results on the subject. Recently, J. Turner [28] has started a studyof nilpotency in the homotopy of simplicial commutative algebras over a field ofcharacteristic 2, with a view towards applications to Quillen’s Conjecture.

1. Differential graded algebras

We use the theory of Eilenberg-Moore derived functors as described in [12, §1,§2].We recall a minimum of material, referring for details to loc. cit .

1.1. Every graded object is concentrated in non-negative degrees, the differentialof every complex has degree −1, and each DG algebra C is graded commutative:

c′c′′ = (−1)|c′||c′′|c′′c′ for all c′, c′′ ∈ C and c2 = 0 for all c ∈ C with |c| odd

where |c| denotes the degree of c. The graded algebra underlying C is denoted C\.We set C [2] = C0+∂(C1)C>1+(C>1)

2 and ind(C) = C/C[2]. This is a complex ofH0(C)-modules and every morphism of DG algebras γ : C → D induces a morphismof complexes of H0(D)-modules ind(γ) : ind(C)⊗H0(C) H0(D)→ ind(D).

1.2. A morphism γ : C → C′ of DG algebras is a quasiisomorphism if it induces anisomorphism in homology; this is often signaled by the appearance of the symbol' next to its arrow. Let C → E be a morphism of DG algebras, such that theC\-module E\ is flat. If γ is a quasiisomorphism, then so is γ⊗C E : E → C′⊗C E.If ε : E → E′ is a quasiisomorphism and the graded C\-module E′\ is flat as well,then C′ ⊗C ε : C′ ⊗C E → C′ ⊗C E′ is a quasiisomorphism.

1.3. A semifree extension of C is a DG algebra C[X ] such that C[X ]\ is isomor-phic to the tensor product over Z of C\ with the symmetric algebra of a free Z-module with basis

⊔i>0X2i and the exterior algebra of a free Z-module with basis⊔

i>0X2i+1; the differential of C[X ] extends that of C.

A semifree Γ -extension of C is a DG algebra C〈X ′〉 such that C〈X ′〉 \ is isomor-phic to the tensor product over Z of C\ with the symmetric algebra of a free Z-module with basisX ′

0, the exterior algebra of a free Z-module with basis⊔i>0X

′2i+1

and the divided powers algebra of a free Z-module with basis⊔i>1X

′2i; the differen-

tial of C〈X ′〉 extends that of C, and for every x′ ∈ X ′2i with i ≥ 1 the jth divided

power x′(j) satisfies ∂(x′(j)) = ∂(x′)x′(j−1) for all j ≥ 1.

1.4. Any morphism of DG algebras C → D factors as the canonical injectionC ↪→ C[X ] followed by a surjective quasiisomorphism C[X ] � D. If ρ : F → D′ isa surjective quasiisomorphism, then for each commutative diagram

C� � //

γ

��

C[X ] //

δ

��

D

δ

��C′ // F

'

ρ// // D′

of morphisms of DG algebras displayed by solid arrows there exists a unique up toC-linear homotopy morphism δ preserving commutativity.


1.5. The diagramsD ← C → E of DG algebras are the objects of a category, whosemorphisms are commutative diagrams of DG algebras

D oo

δ

��

C //

γ

��

E

ε

��D′ oo C′ // E′

In view of 1.4, TorC• (D,E) = H(C[X ] ⊗C E) and Torγ• (δ, ε) = H(δ ⊗γ ε) define afunctor from this category to that of graded algebras. A fundamental property ofthis functor is: If γ, δ, ε above are quasiisomorphisms, then Torγ

•(δ, ε) is bijective.

By 1.2, each factorization C → F'−→ D with F \ flat over C\ yields a unique

isomorphism TorC• (D,E)→ H(F ⊗C E) of graded algebras.

1.6. A DG Γ -algebra is a DG algebra K in which a sequence {x(j) ∈ Kjn}j>0 ofdivided powers is defined for each x ∈ Kn with n even positive, and satisfies alist of standard identities; it can be found in full, say, in [19, (1.7.1), (1.8.1)]. Amorphism of DG Γ -algebras κ : K → L is a morphism of DG algebras such thatκ(x(j)) = (κ(x))(j) for all x ∈ K with |x| even positive and all j ∈ N.

Let K(2) denote the K0-submodule of K generated by K [2] and all x(j), where|x| is even positive and j ≥ 2. Set Γ-ind(K) = K/K(2). This is a complex ofH0(K)-modules. Every morphism of Γ-algebras κ : K → L induces a morphismΓ-ind(κ) : Γ-ind(K)⊗H0(K) H0(L)→ Γ-ind(L) of complexes of H0(L)-modules.

1.7. If K is a DG Γ-algebra, then K〈X ′〉 has a unique structure of DG Γ-algebraextending that of K and preserving the divided powers of the variables x′ ∈ X ′

2i

with i > 0. Every morphism of DG Γ -algebras κ : K → L can be factored as K ↪→K〈X ′〉 � L with second map a surjective quasiisomorphism of DG Γ-algebras. Ifζ : M → L′ is a surjective quasiisomorphism, then for each commutative diagram

K� � //

κ

��

K〈X ′〉 //

λ

��

L

λ��

K ′ // M'

ζ// // L′

of morphisms of DG algebras displayed by solid arrows there exists a unique up toK-linear homotopy morphism of DG Γ-algebras λ making both squares commute.

1.8. Divided powers of a cycle are cycles, but divided powers of a boundary neednot be boundaries. If they are, then the DG Γ-algebra K is called admissible, andH(K) inherits from K a structure of Γ-algebra. This notion of admissibility is lessrestrictive than the one adopted in [12], and lacks some of the desirable propertiesthe latter posesses, but it suffices for the needs of this paper.

Let K � k be a surjective morphism of DG algebras, where k is a field concen-trated in degree 0, and let k ↪→ l be a field extension. If K ↪→ K〈X ′〉� k is a fac-torization as in 1.7, then the unique DG Γ-algebra structure on l〈X ′〉 = K〈X ′〉⊗K lis admissible, cf. [12, (2.6)] or [14, (3.4)]. Thus, Tor defines a functor from the cat-egory of diagrams K � k ↪→ l, with the obvious morphisms, to the category ofΓ-algebras and their morphisms.


2. Factorizations of local homomorphisms

Let ϕ : (R,m, k) → (S, n, l) be a homomorphism of local rings, which is local inthe sense that ϕ(m) ⊆ n. A regular factorization of ϕ is a commutative diagram

R′

##HHHHHHHHH

Rϕ //

;;vvvvvvvvvS

of local homomorphisms such that the R-module R′ is flat, the ring R′/mR′ isregular, and the map R′ → S is surjective.

Regular factorizations are often easily found, for instance, when ϕ is essentiallyof finite type (in particular, surjective), or when ϕ is the canonical embedding of Rin its completion with respect to the maximal ideal. In this paper they are mostlyused through the following construction of Avramov, Foxby, and B. Herzog [11].

2.1. If ϕ : R→ S is the composition of ϕ with the canonical inclusion S → S, then

by [11, (1.1)], ϕ has a regular factorization Rϕ−→ R′ ϕ′

−→ S with a complete localring R′; it is called a Cohen factorization of ϕ. By [11, (1.5)], it can be chosento satisfy the additional condition edimR′/mR′ = edimS/mS; we say that such aCohen factorization is reduced (it is called minimal in [11]). Clearly, any regularfactorization ϕ = π ◦ ι gives rise to a Cohen factorization ϕ = π ◦ ι.

Cohen factorizations need not be isomorphic. However, if Rϕ−→ R′′ ϕ′′

−−→ S also isa Cohen factorization of ϕ, then by [11, (1.2)] there exists a commutative diagram

R′

ϕ′

##HHHHHH

HHHH

R

ϕ

;;vvvvvvvvvv //

ϕ ##HHHH

HHHH

HH R′′′

OO

//

��

S

R′′

ϕ′′

;;vvvvvvvvvv

of local homomorphisms, where the horizontal row is a Cohen factorization, andthe vertical maps are surjections with kernels generated by regular sequences whoseimages in R′′′/mR′′′ can be completed to regular systems of parameters.

2.2. Let (A, p, k) be a local ring. We say that a semifree extension A[X ] hasdecomposable differential if X = X>1 and

∂(X) ⊆ pA[X ] + (X)2A[X ] .

When this condition holds, for each n ≥ 1 there are equalities

Hn(A[X ]/(p, X<n)) = Zn(A[X ]/(p, X<n)) = kXn .

2.3. Let ϕ : (R,m, k)→ (S, n, l) be a local homomorphism.

A minimal model of ϕ is a diagram Reι−→ R′[U ]

eϕ−→ S where the differential

of R′[U ] is decomposable, ϕ is a quasiisomorphism, and ϕ = ϕ0 ◦ ι0 is a regular

factorization. If ϕ has a regular factorization (in particular, if S = S), then ϕ hasa minimal model: The DG algebra R′[U ] is obtained by successively adjoining toR′ sets of variables Un of degree n ≥ 1, so that ∂(U1) minimally generates Ker(π)and ∂(Un) is a minimal set of generators for Hn−1(R[U<n]), cf. [9, (2.1.10)].


The next proposition elaborates on [10, (3.1)].

2.4. Proposition. Let ϕ : (R,m, k) → (S, n, l) be a local homomorphism and let

R→ R′[U ′]→ S and R→ R′′[U ′′]→ S be minimal models of ϕ.For each integer n ≥ 2 there are equalities

card(U ′1)− edim(R′/mR′) = card(U ′′

1 )− edim(R′′/mR′′) ,

card(U ′n) = card(U ′′

n ) ,

and there exist isomorphisms of DG algebras over the field l

l[U ′>n] = R′[U ′]/(m′, U ′

<n)∼= R′′[U ′′]/(m′′, U ′′

<n) = l[U ′′>n] .

Proof. By 2.1 we may assume there is a surjection R′′ → R′ with kernel generatedby a regular sequence x that extends to a minimal generating set of the maximalideal m′′ of R′′. Changing U ′′

1 if need be, we may assume that U ′′ = V t U with∂(V ) = x. The canonical map R′′[V ] → R′ is a quasiisomorphism, R′′[V ] is a DGsubalgebra of R′′[U ′′] and the R′′[V ]\-module R′′[U ′′]\ is free, so the induced map

R′′[U ′′]→ R′′[U ′′]/(V, ∂(V )) = R′[U ]

is a quasiisomorphism, cf. 1.2. Thus, H(R′[U ]) ∼= S, and the differential of R′[U ] isdecomposable because it is induced by that of R′′[U ′′]. By [9, (7.2.3)] there existsan isomorphism R′[U ′] ∼= R′[U ] of DG algebras over R′, so we get

R′[U ′]/(m′, U ′<n)∼= R′[U ]/(m′, U<n)

for all n ≥ 1. The algebra on the right is equal to R′′[U ′′]/(m′′, U ′′<n) for n ≥ 2, so

we have proved the last assertion. In view of 2.2, it implies

lU ′n = Hn(R

′[U ′]/(m′, U ′<n))

∼= Hn(R′[U ]/(m′, U<n)) = lUn .

Thus, we obtain numerical equalities

card(U ′n) = card(Un) = card(U ′′

n ) for n ≥ 2 ;

card(U ′1) = card(U1) = card(U ′′

1 )− card(V )

= card(U ′′1 )−

(edim(R′′/mR′′)− edim(R′/mR′)

).

All the assertions of the proposition have now been established. �

2.5. Definition. Let ϕ : (R,m, k) → S be a local homomorphism, and let R →

R′[U ]→ S be a minimal model of ϕ. The nth deviation of ϕ is the number

εn(ϕ) =

{card(U1)− edim(R′/mR′) + edim(S/mS) for n = 2 ;

card(Un−1) for n ≥ 3 .

By Proposition 2.4, these are invariants of ϕ. Deviations were defined in [10, §3]with a typo in the expression for ε2(ϕ), which is corrected above.

Note that εn(ϕ) ≥ 0 for all n: this is clear for n ≥ 3; for n = 2, use the equalities

card(U1) = rankl

(Ker(ϕ′)

m′ Ker(ϕ′)

);

edim(R′/mR′)− edim(S/mS) = rankl

(Ker(ϕ′)

Ker(ϕ′) ∩ (m′2 + mR′)

).

Vanishing of deviations is linked to the structure of ϕ. We reproduce [10, (3.2)]:

2.6. Proposition. If ϕ : (R,m, k)→ S is a local homomorphism, then the followingconditions are equivalent.


(i) ϕ is flat and S/mS is regular.(ii) εn(ϕ) = 0 for all n ≥ 2 .(iii) ε2(ϕ) = 0 .

Proof. (i) =⇒ (ii) The diagram R → S = S is a Cohen factorization of ϕ, so ϕhas a minimal model with U = ∅.

(iii) =⇒ (i) Choose a reduced Cohen factorization. By definition, ε2(ϕ) = 0

entails U1 = ∅, so S = H0(R′[U ]) = R′, hence S is flat over R and S/mS is regular;

these properties descend to S and S/mS. �

The following notion is basic for the rest of the paper.

2.7. Definition. A local homomorphism ϕ : R → (S, n, l) is complete intersection

(or c.i.) at n, if in some Cohen factorization R → R′ ϕ′

−→ S of ϕ the ideal Ker(ϕ′)is generated by an R′-regular sequence.

Other definitions of c.i. homomorphisms require additional hypotheses on ϕ;when they hold, the general concept specializes properly, cf. [10, (5.2), (5.3)]. Thenext proposition amplifies [10, (3.3)]; it shows, in particular, that the c.i. propertyis detected by every Cohen factorization.

2.8. Proposition. If ϕ : R→ (S, n, l) is a local homomorphism, then the followingconditions are equivalent.

(i) ϕ is complete intersection at n .(ii) εn(ϕ) = 0 for all n ≥ 3 .(iii) ε3(ϕ) = 0 .

Proof. In any minimal model R → R′[U ] → S of ϕ the DG algebra R′[U1] isthe Koszul complex on a minimal set of generators of Ker(ϕ′). If (i) holds, thenU = U1, so (i) implies (ii). If (iii) holds, then H1(R

′[U1]) = 0, so the ideal Ker(ϕ′)is generated by a regular sequence. �

3. Indecomposables

In this section we analyze the divided powers in Tor.

3.1. If (R,m, k) is a local ring, then TorR•(k, k) is a Γ-algebra, cf. 1.8.

Using the functor Γ-ind(−) of Γ-indecomposables defined in 1.6, we set

π•(R) = Γ-ind(TorR• (k, k))

If ϕ : k � l is a field extension, then the canonical isomorphism

TorR•(k, k)⊗k l ∼= TorR

•(k, l)

is one of Γ-algebras, and so induces an isomorphism of graded l-vector spaces

π•(R)⊗k l ∼= Γ-ind(TorR•(k, l))

that we use as identification. Thus, every local homomorphism ϕ : R → (S, n, l)defines an l-linear homomorphism of graded vector spaces

π•(ϕ) : π•(R)⊗k lΓ-ind(Torϕ

• (ϕ,l))−−−−−−−−−−→ π•(S) .


3.2. Example. Let (R,m, k) be a local ring. An acyclic closure of k is a factor-ization R → R〈X ′〉 → k of the epimorphism R → k, as in 1.7, constructed so that∂(X ′

1) minimally generates m and ∂(X ′n) minimally generates Hn−1(R〈X ′

<n〉) foreach n ≥ 2, cf. [9, (6.3)]. By an important theorem of Gulliksen [18] and Schoeller[26], in this case ∂(R〈X ′〉) ⊆ mR〈X ′〉, cf. also [9, (6.3.4)]. This yields isomorphisms

πn(R) ∼= kX ′n for all n ∈ Z .

The nth deviation of R is the number εn(R) = cardX ′n. They measure the

singularity of R: εn(R) = 0 for all n ≥ 2 if and only if ε2(R) = 0, if and only if Ris regular; εn(R) = 0 for all n ≥ 3 if and only if ε3(R) = 0, if and only if R is c.i.,cf. [19, Ch. III], [9, §7]. These results can be derived from Propositions 2.6 and 2.8,since by [9, (7.2.5)] deviations of rings and of homomorphisms are linked as follows:

3.3. If ϕ : A→ R is a surjective local homomorphism with A regular, then

εn(ϕ) = εn(R) for all n ≥ 2 .

The next result is a functorial enhancement of the numerical equality above.

3.4. Theorem. Consider a commutative diagram of morphisms of DG algebras

(1)

Aβ //

ρ

��

� _

��

B

σ

~~

_�

��A[X ]

φ //

eρ '��

B[Y ]

eσ'��

Rϕ // S

where (R,m, k) and (S, n, l) are local rings, (A, p, k) and (B, q, l) are regular localrings, the homomorphisms ϕ and β are local, the homomorphisms ρ and σ aresurjective, Ker(ρ) ⊆ p2 and Ker(σ) ⊆ q2, and the triangles are minimal models.

For each n ≥ 2 there exists a commutative diagram of homomorphisms

πn(R)⊗k l

∼=

��

πn(ϕ) // πn(S)

∼=

��indn−1(l[X ])

indn−1(φ⊗βl) // indn−1(l[Y ])

of l-vector spaces, where the vertical arrows are isomorphisms.

The theorem shows that πn(ϕ) and indn−1(φ⊗β l) determine each other. Theseare very different maps: the first is induced by a morphism of DG Γ-algebras, whiledivided powers have no role in the construction of the second. This accounts forthe intricacies of the proof. In it, and later in the paper, it is convenient to suppressthe effect of TorR1 (k, k) on TorR• (k, k). We do that in a systematic way.

3.5. The reduced torsion algebra of a local ring (R,m, k) is the k-algebra

torR• (k, k) =TorR• (k, k)

TorR• (k, k) · TorR1 (k, k).

Since TorR•(k, k) is a Γ-algebra and the ideal J = TorR

•(k, k) · TorR1 (k, k) is

generated by elements of degree 1, basic properties of divided powers imply thateach element of even degree a ∈ J satisfies a(i) ∈ J for all i ≥ 1. It follows


that torR•(k, k) admits a unique Γ-structure for which the canonical surjection

TorR•(k, k)→ torR

•(k, k) becomes a morphism of Γ-algebras, hence

π>2(R) = Γ-ind(torR• (k, k)) .

If ϕ : R→ (S, n, l) is a local homomorphism, then Torϕ• (ϕ, l) induces a morphism

torϕ•(ϕ, l) : torR

•(k, l)→ torS

•(l, l)

Γ-algebras, so for n ≥ 2 we get commutative diagrams of l-linear homomorphisms

πn(R)⊗k l

∼=

��

πn(ϕ) // πn(S)

∼=

��Γ-indn(torR• (k, l))

Γ-indn(torϕ• (ϕ,l)) // Γ-indn(torS• (l, l))

The proof of Theorem 3.4 takes up the rest of the section. Only its statement isused later, so the reader may skip to the next section without loss of continuity.

We start by forming a diagram of morphisms of DG Γ-algebras

(2)

Aβ //

� _

��

B_�

��A〈X ′

1〉κ //

ε '��

B〈Y ′1〉

η'��

kϕ // l

in the following order. First we form the vertical sides by choosing them to beacyclic closures of the respective residue fields. Next we note that since both A andB are regular local rings, the DG algebras A〈X ′

1〉 and B〈Y ′1〉 are Koszul complexes

on minimal sets of generators of p and q, respectively. Finally, we use 1.7 to choosea morphism κ that preserves the commutativity of the rectangle.

Base change from Diagram (2) yields the central rectangles in the diagram

(3)

R〈X ′〉

ϕ

((R

ϕ //? _oo� _

τ

��

S � � //_�

θ

��

S〈Y ′〉

R〈X ′〉

'$$ $$JJJJJJJJJJR〈X ′

1〉? _oo ϕ⊗βκ //

��

S〈Y ′1〉

� � //

��

S〈Y ′〉

'zzzzuuuuuuuuuu

kϕ // l

of morphisms of DG Γ-algebras. The rest is constructed as follows. In view ofthe hypotheses on ρ and σ, minimal sets of generators of p and q map to minimalsets of generators of m and n, respectively. By Example 3.2 the DG algebrasR〈X ′

1〉 = R⊗AA〈X ′1〉 and S〈Y ′

1〉 = S⊗BB〈Y ′1〉 can be extended to acyclic closures

R ↪→ R〈X ′〉 � k and S ↪→ S〈Y ′〉 � l. Finally, the morphism ϕ is chosen so as topreserve the commutativity of the diagram: this is possible by 1.7.


3.6. Lemma. Diagram (3) induces a commutative diagram

l〈X ′>2〉

∼= // torR•(k, l)

torϕ• (ϕ,l) //

∼=

��

torS•(l, l)

∼=

��

l〈Y ′>2〉

∼=oo

TorR〈X′1〉

•(k, l)

Torϕ⊗βκ

• (ϕ,l) // TorS〈Y′1 〉

•(l, l)

of homomorphisms of Γ -algebras.

Proof. By construction, R〈X ′〉 and S〈Y ′〉 are acyclic closures. In view of 3.2, thismeans that there are inclusions ∂(R〈X ′〉) ⊆ mR〈X ′〉 and ∂(S〈Y ′〉) ⊆ nS′〈Y ′〉.These inclusions provide the equalities in the commutative diagram

l〈X ′〉

��

ϕ⊗ϕl

))TorR• (k, l)

Torϕ• (ϕ,l) //

Torτ•(k,l)

��

TorS• (l, l)

Torθ•(l,l)

��

l〈Y ′〉

��l〈X ′

>2〉 TorR〈X′1〉

• (k, l)Tor

ϕ⊗βκ

• (ϕ,l) // TorS〈Y′1 〉

• (l, l) l〈Y ′>2〉

induced by Diagram (3). By 1.8, all the maps are morphisms of Γ-algebras.The inclusions noted above also show that the external vertical maps are the

canonical surjections of graded algebras, whose kernels are the ideals generated byX ′

1 and Y ′1 respectively. It follows that Ker(Torτ

•(k, l)) is generated by TorR1 (k, l),

and Ker(Torθ•(l, l)) is generated by TorS1 (l, l). In view of the definition of the reducedTor functor in 3.5, the diagram above induces the desired diagram. �

We refine Diagram (1) to a commutative diagram of morphisms of DG algebras

(4)

A[X ]φ //

eρ

' && &&LLLLLLLLLLL

κ

��::

::

::

::

B[Y ]

eσ

'xxxxrrrrrrrrrrr

λ

��

��

��

��

�

Rϕ // S

A〈V 〉 Φ //

ρ'

OOOO

B〈W 〉

σ '

OOOO

A?�

OO

β //+ �

99sssssssssssB� ?

OO

3 S

eeKKKKKKKKKKK

by performing the following steps. First we invoke 1.7 to construct factorizations

A ↪→ A〈V 〉ρ−→ R of ρ and B ↪→ B〈W 〉

σ−→ S of σ. Next we choose by 1.7 a morphism

of DG Γ-algebras Φ so as to preserve the commutativity of the already constructedpart of the diagram. Finally, we use 1.4 to obtain morphisms of DG algebras κ andλ which preserve the commutativity of the lateral trapezoids.


It should be noted at this point that, in general, Φκ 6= λφ. Using Diagrams (2)and (4) we produce a diagram of morphisms of DG algebras

(5)

A[X ]〈X ′1〉

φ⊗βκ //

eρ⊗AA〈X′1〉

'

&& &&MMMMMMMMMMMMMMMM

κ⊗AA〈X′1〉

'

��<<

<<

<<

<<

<<

<<

A[X]⊗Aε'

��

B[Y ]〈Y ′1〉

eσ⊗BB〈Y ′1〉

'

xxxxqqqqqqqqqqqqqqq

λ⊗BB〈Y ′1 〉

'

��

��

��

��

��

��

B[Y ]⊗Bη '

��

R〈X ′1〉

ϕ⊗βκ // S〈Y ′1〉

A〈V 〉〈X ′1〉

ρ⊗AA〈X′1〉'

OOOO

Φ⊗βκ

//

A〈V 〉⊗Aε '

��

B〈W 〉〈Y ′1〉

σ⊗BB〈Y ′1〉 '

OOOO

B〈W 〉 ⊗Bη'

��k〈V 〉

Φ⊗βϕ // l〈W 〉

k[X ]

κ⊗Ak

'

88qq

qq

qq

qq φ⊗βϕ // l[Y ]

λ⊗B l

'

ffMM

MM

MM

MM

where the central rectangles are formed by morphisms of DG Γ-algebras, allnon-horizontal arrows are quasiisomorphisms due to 1.2, and almost all pathscommute—the possible exception being the paths around the two trapezoids withhorizontal bases and hyphenated sides.

From 1.5 and 1.8 we deduce the following result.

3.7. Lemma. The maps in Diagram (5) induce a commutative diagram

TorR〈X′1〉

•(k, l)

∼=

��

Torϕ⊗βκ

• (ϕ,l) // TorS〈Y′1 〉

•(l, l)

∼=

��Tork〈V 〉

• (k, l)Tor

Φ⊗βϕ

• (ϕ,l) // Torl〈W 〉• (l, l)

of homomorphisms of Γ -algebras, where the vertical maps are isomorphisms. �

We pause to recall some classical material on bar-constructions.

3.8. Let C be a connected DG algebra over the field k, which means that C0 = kand ∂(C1) = 0. The bar construction (Bk(C), ∂) is a connected DG Γ-algebra overk, with multiplication (called shuffle product) and divided powers constructed in[15, Exp. 7, §1)]; cf. also [24, Ch. X, §12]. It has a basis consisting of symbols[c1|c2| · · · |cp] of degree p+ |c1|+ · · ·+ |cp|, where the ci range independently over abasis of C>1 and p = 0, 1, 2 . . . . The element [c1|c2| · · · |cp] has weight p; the weight

of x · y is the sum of those of x and y; if |x| is even positive, then the weight of x(i)

is i times that of x. In general, the DG Γ-algebra Bk(C) is not admissible.There exists a DG algebra (Bk(C), ∂) such that Bk(C)\ = C\⊗kBk(C)\ as graded

algebras, ∂ extends the differential of C, the isomorphism Bk(C) ⊗C k ∼= Bk(C) isone of DG algebras, and the augmentation Bk(C) → k is a quasiisomorphism of


DG algebras. If C is a DG Γ-algebra, then by [15, Exp. 7, §5)] so is Bk(C), themap Bk(C)→ Bk(C) is a morphism of DG Γ-algebras, and Bk(C) is admissible.

The bar construction is natural for morphisms γ : C → C′ of connected DGk-algebras; a morphism of DG Γ-algebras Bk(γ) : Bk(C)→ Bk(C

′) is given by

(6) Bk(γ)([c1| · · · |cp]) = [γ(c1)| · · · |γ(cp)]

There is a canonical isomorphism Bk(C)⊗k l ∼= Bl(C⊗k l) of DG Γ-algebras overl. In conjunction with 1.5, it induces isomorphisms of graded algebras

TorC•

(k, l) ∼= H(Bk(C)⊗C l) = H(Bk(C)⊗k l) ∼= H(Bl(C ⊗k l))

which are natural with respect to morphisms of connected DG k-algebras. WhenC is a DG Γ-algebra the isomorphisms above are of Γ-algebras, cf. 1.8.

3.9. Lemma. The DG Γ -algebras Bl(l[X ]) and Bl(l[Y ]) are admissible, and themaps in Diagram (5) induce a commutative diagram

Tork〈V 〉• (k, l)

TorΦ⊗βϕ

• (ϕ,l) //

∼=

��

Torl〈W 〉• (l, l)

∼=

��HBl(l[X ])

HBl(φ⊗βl) // HBl(l[Y ])

of morphisms of Γ -algebras over l, where the vertical maps are isomorphisms.

Proof. Diagram (5) induces a diagram of homomorphisms of graded algebras

TorA[X]〈X′1〉

•(k, l) //

∼=

$$JJJJJJJJJJJJJJ

∼=

��777

7777

7777

7777

7777

7777

TorA[X]⊗Aε• (k,l)∼=

��

TorB[Y ]〈Y ′1 〉

•(l, l)

∼=

zzuuuuuuuuuuuuu

∼=

��

��

��

��

��

��

∼=

��

TorR〈X′1〉

•(k, l)

Torϕ⊗βκ

• (ϕ,l)// TorS〈Y′1 〉

•(l, l)

TorA〈V 〉〈X′1〉

• (k, l)

∼=

OO

//

∼=

��

TorB〈W 〉〈Y ′1 〉

• (l, l)

Torσ⊗BB〈Y ′

1〉• (l,l)

∼=

OO

∼=

��Tork〈V 〉

•(k, l)

TorΦ⊗β ϕ

• (ϕ,l)

// Torl〈W 〉•

(l, l)

Tork[X]•

(k, l)Tor

φ⊗βϕ

• (ϕ,l) //

Torκ⊗Ak• (k,l)

∼=

99tttttttttttttt

Torl[Y ]•

(l, l)

Torλ⊗Bl• (l,l)

∼=

ddIIIIIIIIIIIII

where the non-horizontal maps are bijective by 1.5. All paths commute, ex-cept possibly those around two trapezoid with horizontal bases—the one on thefloor and the larger of the pair at the ceiling. Composing either path from

TorA[X]〈X′1〉

• (k, l) to TorB〈W 〉〈Y ′1〉

• (l, l) with the isomorphism Torσ⊗BB〈Y ′1 〉

• (l, l) weget the same map, so the upper trapezoid commutes. Using this, we see that


the isomorphism TorA[X]⊗Aε•

(k, l), composed with either path from Tork[X]•

(k, l) to

Torl〈W 〉• (l, l), yields the same map, so the lower trapezoid commutes as well.

We inflate this trapezoid to a diagram of homomorphisms of graded algebras

Tork〈V 〉• (k, l)

TorΦ⊗βϕ

• (ϕ,l) //

∼=

��

Torl〈W 〉• (l, l)

∼=

��HBl(l〈V 〉)

HBl(Φ⊗βϕ) // HBl(l〈W 〉)

HBl(l[X ])HBl(φ⊗βϕ)

//

HBl(κ⊗Al)

OO

HBl(l[Y ])

HBl(λ⊗B l)

OO

Tork[X]•

(k, l)Tor

φ⊗βϕ

• (ϕ,l) //

Torκ⊗Ak• (k,l)

∼=

<<

∼=

66nnnnnnnnnnnn

Torl[Y ]•

(l, l)

Torλ⊗Bl• (l,l)

∼=

aa

∼=

ggOOOOOOOOOOOO

From 3.8 we know that the maps pointing inward from the corners are bijective,and that the upper rectangle, both triangles, and the inner trapezoid commute. Weconclude that the lower rectangle commutes and its vertical arrows are bijective.

Referring to 3.8 again, we note that all maps in the lower rectangle are inducedby morphisms of DG Γ-algebras, and that Bl(l〈V 〉) and Bl(l〈W 〉) are admissible.As Bl(κ ⊗A l) and Bl(λ ⊗B l) are quasiisomorphisms, it follows that the DG Γ-algebras Bl(l[X ]) and Bl(l[Y ]) are admissible and the maps in the lower rectangleare isomorphisms of Γ-algebras. To finish the proof we remark, with a final referenceto 3.8, that the upper rectangle is formed by homomorphisms of Γ-algebras. �

3.10. Let C be a connected DG algebra over l. The differential ∂ of the bar con-struction Bl(C) has the form ∂′ + ∂′′, where

∂′([c1|c2| · · · |cp]) =

p−1∑

j=1

(−1)|c1|+···+|cj|+j [c1| · · · |cjcj+1| · · · |cp]

∂′′([c1|c2| · · · |cp]) =

p∑

j=1

(−1)|c1|+···+|cj−1|+j [c1| · · · |∂(cj)| · · · |cp]

Thus, the l-span Fq(C) of the elements [c1| · · · |cp] of degree at most (p + q) for

p = 0, 1, 2 . . . , is a subcomplex of Bl(C). The page 0E of the spectral sequence ofthe filtration {Fq(C)} is a complex of graded vector spaces with 0d induced by ∂′. Itcan also be obtained by tensoring with l over C\ the complex of graded C\-modules

(7)

· · · −→ C\ ⊗l Σp(C\

>1⊗p)

δp−→ C\ ⊗l Σ

p−1(C\>1

⊗(p−1)) −→ · · ·

δ(Σp(c0 ⊗ c1 ⊗ · · · ⊗ cp)) =

p−1∑

j=0

(−1)jΣp−1(c0 ⊗ · · · ⊗ cjcj+1 ⊗ · · · ⊗ cp)

where for a graded vector space M we let ΣpM denote the graded space with(ΣpM)n = Mn−p for all n, and Σ

p : M → ΣpM be the degree p bijection definedby the maps idMn

. The complex (7) is the standard resolution of l by free gradedC\-modules, so the spectral sequence of the filtration {Fq(C)} has

(8) 1Ep,q = TorC\

p (l, l)q =⇒ HBl(C)


In particular, the following equalities hold:

1Ep,q =

{0 for p ≤ 0 and all q except for (p, q) = (0, 0) ;

indq(C) for p = 1 and all q .

The differentials of the spectral sequence act according to the pattern

rdp,q : rEp,q −→rEp+r−1,q−r for each r ≥ 0

so for every q ≥ 1 at the edge p = 1 the spectral sequence defines l-linear maps

HBl(C)n �∞E1,n−1 � · · ·� 1E1,n−1 = indn−1(C) .

where the kernel of the first one is the image of Hn(F2(C)) → HBl(C)n. Shuffle

products and divided powers in Bk(C) are homogeneous with respect both to degreeand to weight, cf. 3.8, so the subspace Bl(C)(2) of 1.6 is contained in F2(C). Thus,if Bl(C) is admissible, then for each n ≥ 1 the maps above define a composition

(9) νCn : Γ-indn(HBl(C)) �∞E1,n−1 � · · ·� 1E1,n−1 = indn−1(C)

of l-linear homomorphisms. Formula (6) yields inclusions Bl(γ)(Fq(C)) ⊆ Fq(C′)

for all q and every morphism γ : C → C′ of connected DG algebras over l. It followsthat the spectral sequence (8) above is natural with respect to such morphisms, andhence so are its edge homomorphism νCn .

Proof of Theorem 3.4. For each n ≥ 2 we form a commutative diagram

πn(R)⊗k lπn(ϕ) // πn(S)

lX ′n

∼= // Γ-indn(torR•(k, l))

Γ-indn(torϕ• (ϕ,l)) //

∼=

��

Γ-indn(torS•(l, l))

∼=

��

lY ′n

∼=oo

Γ-indn(HBl(l[X ]))Γ-indn(HBl(φ⊗βl)) // Γ-indn(HBl(l[Y ]))

indn−1(l[X ]))indn−1(φ⊗βl) //

��νl[X]

n

indn−1(l[Y ])��νl[Y ]

n

of l-vector spaces as follows: The top rectangle comes from 3.5. The middle part isobtained by stacking the commutative diagrams of Lemmas 3.6, 3.7, and 3.9, thentaking Γ-indecomposables, as in 1.6. The bottom rectangle reflects the naturalityof the edge homomorphisms νn defined in (9).

To finish the proof we show that its vertical maps are bijective. It suffices to do

this for νl[X]n . By the isomorphisms above and 3.3, its source and target have the

same rank, so it is enough to prove that it is surjective. To this end we analyze thespectral sequence (8). A well known computation, cf. e.g. [9, (7.2.9)], gives

1Ep,q = Torl[X]\

p (l, l)q ∼= l〈X ′′〉p,q where cardX ′′m,n =

{cardXn if m = 1 ;

0 otherwise .

From 3.3 we know that card(Xn) = card(X ′n+1) for n ≥ 1, so the graded vector

space associated with the bigraded space 1E is isomorphic to l〈X ′>2〉. By Lemmas

3.6, 3.7, and 3.9 the latter space is isomorphic to HBl(l[X ]). This is the abutment of


the spectral sequence (8), so it is isomorphic to the graded vector space associatedwith the bigraded space ∞E. Putting these remarks together, for each n we get

∑

p+q=n

rankl1Ep,q = rankl(l〈X

′>2〉n) = rankl Hn(Bl(l[X ])) =

∑

p+q=n

rankl∞Ep,q .

They imply that the spectral sequence (8) stops on the page 1E, so in the decom-position (9) the injections r+1E1,n−1 �

rE1,n−1 are bijective for all n ≥ 1 and

1 ≤ r ≤ ∞. As a consequence, the map νl[X]n is surjective, as desired. �

4. Almost small local homomorphisms

We introduce a class of maps of major importance for this paper.

4.1. Definition. A local homomorphism ϕ : (R,m, k)→ (S, n, l) is said to be almost

small if the kernel of the homomorphism Torϕ• (ϕ, l) : TorR• (k, l) → TorS• (l, l) ofgraded algebras is generated by elements of degree 1.

The name reflects the relation of the new concept to that of small homomor-phism, defined in [8] by the condition that the map Torϕ• (ϕ, l) is injective.

4.2. Example. It is proved in [8, (4.1)] that for every ring (R,m, k), and for eachideal a contained in ms for some sufficiently large s, the canonical epimorphismR→ R/a is small. An effective bound on s has been found recently by Liana Sega.

Namely, let G be the symmetric algebra of the k-vector space m/m2, let grm

(R)be the associated graded ring of R, and extend the identity map of m/m2 to a homo-morphism of graded k-algebras G→ gr

m(R). Let pol regR denote the Castelnuovo-

Mumford regularity of the graded G-module grm(R), that is

pol regR = supi∈N

{j ∈ Z

∣∣ TorGi (grm

(R), k)i+j 6= 0}.

By [27, (6.2)] the epimorphism R→ R/ms is Golod for all s ≥ 2+pol regR. Golodhomomorphisms are small by [8, (3.5)], so the factorization R→ R/a→ R/ms andfunctoriality imply that R→ R/a is small for every ideal a contained in ms.

By [8, (3.1)], ϕ is small if and only if π•(ϕ) is injective. We characterize almostsmallness in similar terms, and by means of reduced Tor-algebras, cf. 3.5.

4.3. Proposition. Let ϕ : (R,m, k)→ (S, n, l) be a local homomorphism.The following conditions are equivalent.

(i) ϕ is almost small.(ii) π>2(ϕ) is injective.(iii) torϕ• (ϕ, l) is injective.

Proof. Using the fact that Torϕ•(ϕ, l) is a homomorphism of Hopf Γ -algebras,

it is proved in [8, (1.3)] that there exists a subset G ⊂ TorR•(k, l) such that

Ker(Torϕ•(ϕ, l)) = (l〈G〉)>1 TorR

•(k, l) and the following hold:

(1) The image of G in π•(R)⊗k l is a basis of Ker(π•(ϕ)).

(2) The graded l〈G〉-module TorR•(k, l) is free.

Clearly, (i) ⇐⇒ (ii) follows from (1). The freeness of TorR• (k, l) over l〈TorR1 (k, l)〉,that of TorS

•(l, l) over l〈TorS1 (l, l)〉, and (2) yield (i) ⇐⇒ (iii). �

Vanishing of π>2(R) characterizes regularity, cf. Example 3.2, so we get

4.4. Corollary. If the ring R is regular, then ϕ is almost small. Conversely, if thecanonical surjection ε : R→ k is almost small, then R is regular. �


Using the functoriality of π•( ), we see that the proposition also implies

4.5. Corollary. Let ψ : Q→ R and ϕ : R→ S be local homomorphisms.

(a) If ψ and ϕ are almost small, then ϕ ◦ ψ is almost small.(b) If ϕ ◦ ψ is almost small, then ψ is almost small.(c) If ϕ ◦ ψ is almost small and π>2(ψ) is bijective, then ϕ is almost small. �

As a further corollary, we get another example of almost small homomorphisms.

4.6. Example. If ϕ is flat and char(l) = 2, then ϕ is almost small by Andre [5].

Here is what is known for flat homomorphisms in general.

4.7. Remark. If ϕ : (R,m, k)→ (S, n, l) is a flat local homomorphism, then by [7,(1.1)] for every i ≥ 1 there exists an exact sequence of l-vector spaces

0 // π2i(R)⊗k lπ2i(ϕ) // π2i(S) // π2i(S/mS)

ð2i //

// π2i−1(R)⊗k lπ2i−1(ϕ) // π2i−1(S) // π2i−1(S/mS) // 0

Andre [6] proved∑∞i=1 rankl(ð2i) ≤ edim(S/mS)− depth(S/mS) and conjectured

that π2i−1(ϕ) is injective for all i ≥ 2. In view of Proposition 4.3, the conjecturecan be restated to say that every flat homomorphism is almost small.

The next proposition fails for small homomorphisms, and presents one of themain technical reasons for working with almost small homomorphisms.

4.8. Proposition. Let ϕ : (R,m, k)→ (S, n, l) be a local homomorphism.

The following maps are almost small simultaneously: ϕ, ϕ : R → S, ϕ : R → S,

and ρ : R′ → S, where Rϕ−→ R′ ρ

−→ S is a regular factorization of ϕ.

Proof. The map ϕ : R→ S is the composition of ϕ with the completion map S → S,

and also the composition of the completion map R → R with ϕ. As π•( ) appliedto either completion map yields an isomorphism, Corollary 4.5 shows that ϕ, ϕ,and ϕ are almost small simultaneously. Finally, π>2(R

′/mR′) = 0 because R′/mR′

is regular, cf. Example 3.2, so π>2(ϕ) is bijective by the exact sequence of Remark4.7. Thus, ϕ and ϕ are almost small simultaneously by Corollary 4.5.3. �

As an application, we show how to obtain almost small homomorphisms byfactoring complete intersection homomorphisms.

4.9. Corollary. Let ϕ : R → (S, n, l) be a local homomorphism. If there exists alocal homomorphism ξ : S → (S′, n′, l′) such that ξ ◦ϕ is c.i. at n′, then ϕ is almostsmall. In particular, if ϕ is c.i. at n, then it is almost small.

Proof. Let R → R′ ϕ′

−→ S′ be a reduced Cohen factorization of the composition

ξ ◦ ϕ : R → S′. By hypothesis, Ker(ϕ′) is generated by an R′-regular sequence, so[19, (3.4.1)] shows that πn(ϕ

′) is injective for n = 2 and bijective for n ≥ 3. ByProposition 4.3, the map ϕ′ is almost small, which implies, by Proposition 4.8, thatξ ◦ ϕ is almost small as well. It remains to invoke Corollary 4.5.b. �

Extending Example 4.2, we provide a numerical test for almost smallness.

4.10. Proposition. Let ϕ : (R,m, k)→ (S, n, l) be a local homomorphism. If

lengthS(S/ns) =

s−1∑

i=0

(e+ i− 1

i

)lengthR(R/ms−i)


for e = edim(S/mS) and s = 2 + pol regR, then ϕ is almost small.

Proof. First we note that for every integer r there is an inequality

lengthS(S/nr) ≤r−1∑

i=0

(e+ i− 1

i

)lengthR(R/mr−i) .

Indeed, if R → (R′,m′, l)ϕ′

−→ S is a reduced Cohen factorization, then the righthand side of the formula above is equal to lengthR′(R′/m′r). The surjective homo-

morphism ϕ′r : R′/m′r → S/nr induced by R′ → S yields the inequality above, and

ϕ′r is bijective if and only if equality holds, that is, if and only if Ker(ϕ′) ⊆ m′r.By the preceding argument, our hypothesis implies Ker(ϕ′) ⊆ m′s with s =

2 + pol regR. On the other hand, the associated graded rings of R′ and R arelinked by an isomorphism of graded l-algebras

grm

′(R′) ∼= l ⊗k grm

(R)[x1, . . . , xe]

where x1, . . . , xe are indeterminates. It follows that pol reg(R′) = pol reg(R). Ex-ample 4.2 now shows that the homomorphism ϕ′ is small. In view of Proposition4.8, it follows that the homomorphism ϕ is almost small. �

The next result is a structure theorem for morphisms of minimal models overcertain almost small homomorphisms. A key ingredient of the proof is the generalresult on the map π•(ϕ) established in Theorem 3.4.

4.11. Theorem. Let ρ : (A, p, k) → R and ϕ : (R,m, k) → S be surjective homo-morphisms of local rings, with A regular and Ker(ρ) ⊆ p2. If ϕ is almost small,then there exists a commutative diagram of morphisms of DG algebras

A � � //

β

&&� _

��ρ

��

A[U ]eβ

'// //

� _

��

B

σ

��

_�

��A[X ]

� � //

eρ '

��

A[X,U, T ]χ

'// //

υ '��

B[X,T ]

eσ'

��R

� � //

ϕ

99R[U, T ]eϕ

'// // S

where (B, q, k) is a regular local ring, β and σ are surjective homomorphisms,Ker(σ) is contained in q2, U = U1, the external rows and columns are minimalmodels, χ and υ are surjective quasiisomorphisms.

Proof. Choose a subset a ⊂ A mapping to a basis of (Ker(ϕ) + m2)/m2.As a is part of a regular system of parameters for A, the local ring (B, q, k) =

(A/(a), p/(a), k) is regular. Since a is contained in the kernel of ϕ ◦ ρ, this mapfactors as a composition of surjective homomorphisms β : A → B and σ : B → S.The choice of a ensures that Ker(σ) is contained in q2.


Using 2.3, we form a commutative diagram of morphisms of DG algebras

Aβ //

ρ

��

� _

��

B

σ

~~

_�

��A[X ]

φ //

eρ '��

B[Y ]

eσ'��

Rϕ // S

It induces morphisms of DG algebras

φ′ : B[X ] = A[X ]⊗A Bφ⊗AB−−−−→ B[Y ]⊗A B = B[Y ] ;

φ : k[X ] = A[X ]⊗A kφ⊗Ak−−−−→ B[Y ]⊗A k = k[Y ] .

As ϕ is almost small, Theorem 3.4 shows that the k-linear map

ind(φ) : ind(k[X ]) −→ ind(k[Y ])

is injective. Thus, the set ind(φ)(X) is linearly independent in ind(k[Y ]).Choose a subset T in B[Y ] whose image in ind(k[Y ]) extends ind(φ)(X) to a

basis. It follows that T is a set of free variables over k[X ], and so the map φ isinjective. This map is equal to φ′ ⊗B k, and φ′ is a map of graded free B-modules,so we conclude by Nakayama’s Lemma that φ′ is injective and T is a set of freevariables generating B[Y ] over φ′(B[X ]). Changing variables in B[Y ], we replaceB[Y ] by B[X,T ] and φ′ by the canonical inclusion B[X ] ↪→ B[X,T ].

Let A[U ] be the Koszul complex with U = U1 and ∂(U) = a, and set A[X,U ] =A[X ]⊗A A[U ]. These algebras appear in a commutative diagram of DG algebras

A � � //� _

��

A[U ]� _

��

eβ

'// // B

_�

��A[X ] � � // A[X,U ]

� _

��

χ0

'// // B[X ]

_�

��A[X ]

� � // A[X,U, T ]χ

'// // B[X,T ]

where β is the canonical augmentation and χ0 = A[X ]⊗A β. Since a is an A-regular

sequence, β is a quasiisomorphism; by 1.2, χ0 is a quasiisomorphism as well.The map χ is built inductively, starting with χ0. Using the inclusions ∂(T1) ⊆

Ker(σ) ⊆ q2, pick for each t ∈ T1 an element pt ∈ p2 with β(pt) = ∂(t). Let A[T1]be the Koszul complex with ∂(t) = pt and χ1 the morphism of DG algebras

A[X,U, T1] = A[X,U ]⊗A A[T1]χ0⊗AA[T1]−−−−−−−→ B[X ]⊗A A[T1] = B[X,T1] .

By 1.2, χ1 is a surjective quasiisomorphism. Assume next that a surjective quasi-isomorphism χn : A[X,U, T6n] → B[X,T6n] is available for some n ≥ 1. For eacht ∈ Tn+1 we pick a cycle zt ∈ A[X,U, T6n]n such that χn(zt) = ∂(t), then we setA[X,U, T6n+1] = A[X,U, T6n][Tn+1 |∂(t) = zt] and define χn+1 to be the extensionof χn satisfying χn+1(t) = t for all t ∈ Tn+1. It is easy to verify that this map is


a surjective quasiisomorphism, cf. also [9, (7.2.10)]. Taking direct limits, we obtainthe surjective quasiisomorphism χ displayed in the diagram.

The diagram above provides the two upper squares of the diagram in the theorem.Its lower left square is obtained by base change along ρ. For its lower right square,we factor σ ◦χ through υ = ρ⊗A[X] A[X,U, T ] to get a surjection ϕ : R[U, T ]→ S.

The top row and two side columns of the diagram are minimal models by con-struction. Furthermore, χ, υ, and ϕ are surjective quasiisomorphisms: the firstby construction, the second by 1.2, and the third due to the commutativity of thediagram. Since the differential of R[U, T ] is induced by that of A[X,U, T ], to provethat the lower row is a minimal model it suffices to establish that the differentialon A[X,U, T ] is decomposable.

For any y ∈ A[X,U, T ] with |y| = n+ 1 ≥ 2, write ∂(y) in the form

∂(y) =∑

x∈Xn

cxx+∑

u∈Un

buu+∑

t∈Tn

att+ w ∈ A[X,U, T ]

with at, bu, cx ∈ A and w ∈ (X,U, T )2A[X,U, T ]. In the resulting equality

∂(χ(y)) =∑

x∈Xn

β(cx)x+∑

t∈Tn

β(at)t+ χ(w) ∈ B[X,T ]

we have χ(w) ∈ (X,T )2B[X,T ]. The differential of B[X,T ] is decomposable, so forall t ∈ Tn and x ∈ Xn we obtain β(at), β(cx) ∈ q, that is, at, cx ∈ p. Since U = U1,we have bu = 0 unless n = 1. If n = 1, then w = 0, so the equality ∂2(y) = 0 yields

∑

u∈U1

bu∂(u) = −∑

x∈X1

cx∂(x)−∑

t∈T1

at∂(t) .

By construction, we have ∂(x) ∈ Ker(ρ) ⊆ p2 for all x ∈ X1 and ∂(t) = pt ∈ p2 forall t ∈ T1, so the last equality yields

∑u∈U1

bu∂(u) ∈ p2. As ∂(U1) = a is part of aregular system of parameters, this implies bu ∈ p for all u ∈ U1, so the differentialof A[X,U, T ] is decomposable. �

5. Weak category of a local homomorphism

We introduce a notion motivated by Felix and Halperin’s [17, (4.3)] definitionof rational Lusternik-Schnirelmann category cat0(X) of a simply connected CWcomplex X of finite type. Weak category captures a Looking Glass [12] image ofa corollary of the Mapping Theorem: If cat0(X) ≤ s, then by [17, (5.1)] for eachn ≥ 2 the n-connected cover Xn of X satisfies cat0(Xn) ≤ s, hence by [17, (4.10)]the product of any (s+ 1) cohomology classes in H>1(Xn; Q) is equal to 0.

5.1. Definition. If (B, q, k) is a local ring and B[V ] is a semifree extension withdecomposable differential, then we define a notion of weak category by the formula

wcat(B[V ]) = inf

s ∈ N

∣∣∣∣∣∣∣

for each n ≥ 2 the product of any

(s+ 1) elements of positive degree

in H(B[V ]/(q, V<n)) is equal to 0

.

Finite weak category can often be detected by using a variant of [13, (1.2)]:

5.2. Proposition. If (B, q, k) is a local ring, B[V ] a DG algebra with decomposabledifferential, and B[V ]→ S[W ] a surjective morphism of DG algebras, then

wcat(S[W ]) ≤ sup{s ∈ N | Hs(B[V ]/qB[V ]) 6= 0} .


Proof. Set k[V ] = B[V ]/qB[V ] and k[W ] = S[W ]/qS[W ]. All DG algebras underconsideration are images of B[V ], so their differentials are decomposable. We havewcat(S[W ]) = wcat(k[W ]) by definition. As the induced morphism k[V ] → k[W ]is surjective, we get wcat(k[W ]) ≤ sup{s ∈ N | Hs(k[V ]) 6= 0} from [13, (1.2)]. �

5.3. Definition. If ϕ : R → S is a local homomorphism and R → R′[U ] → S is aminimal model of ϕ, then we define the weak category of ϕ by the equality

wcat(ϕ) = wcat(R′[U ]) .

Proposition 2.4 shows that it does not depend on the choice of minimal model.

By [10, §3], the sequence (εn(ϕ)) is positive and grows exponentially when ϕ isnot c.i. and fdR S is finite. A close reading of the proofs shows the last condition isused only to ensure wcat(ϕ) <∞, cf. Theorem 5.7, so at no further expense we get

5.4. Theorem. Let ϕ : R→ (S, n, l) be a local homomorphism.If wcat(ϕ) is finite, then the following conditions are equivalent.

(i) ϕ is not complete intersection at n .(ii) εn(ϕ) > 0 for all n ≥ 2 .

(iii) lim supnn√εn(ϕ) > 1 .

(iv) There exist a real number c > 1 and a sequence of integers sj with

0 < 2sj ≤ sj+1 ≤ (wcat(ϕ) + 1)sj and εsj(ϕ) > csj for all j ≥ 1 .

Proof. Proposition 2.8 shows that (ii) or (iii) implies (i).If (i) holds, then εn(ϕ) 6= 0 for n = 2, 3 by Propositions 2.6 and 2.8, and

εn(ϕ) 6= 0 n ≥ 4 by the proof of [10, (3.4)]; thus, (i) implies (ii).The proof of [10, (3.10)] shows that (i) implies (iv). �

5.5. Corollary. The following conditions are equivalent.

(i) ϕ is complete intersection at n .(ii) wcat(ϕ) <∞ and εn(ϕ) = 0 for some n ≥ 2 .

(iii) wcat(ϕ) <∞ and lim supnn√εn(ϕ) ≤ 1 .

(iv) wcat(ϕ) = 0 .

Proof. The theorem shows that (i) follows from either (ii) or (iii).

Let R → R′[U ] → S be a minimal model of ϕ : R → S. Proposition 2.8 showsthat ϕ is c.i. at n if and only if U = U1. Thus, (i) implies (iv) by definition ofwcat(ϕ). Conversely, if (iv) holds, then U = U1 by 2.2, hence (i) holds.

Finally, conditions (i) and (iv) imply (ii) and (iii) by Proposition 2.8. �

Next we establish a most important property of almost small homomorphisms.

5.6. Theorem. If a local homomorphism ϕ : R→ S is almost small, then

wcat(ϕ) ≤ edimS − depthS .

Proof. Let Rϕ−→ R′ ϕ′

−→ S be a Cohen presentation of ϕ. As R′ =−→ R′ ϕ′

−→ S is aCohen presentation of ϕ′ = ϕ′, we have wcat(ϕ) = wcat(ϕ′). On the other hand,

the map ϕ′ is almost small by Proposition 4.8. Furthermore, edim S = edimS and

depth S = depthS. Thus, we may assume that R and S are complete and that thelocal homomorphism ϕ : (R,m, k)→ (S, n, k) is surjective and almost small.

Choose a regular local ring (A, p, k) and a surjective homomorphism ρ : A → Rwith Ker(ρ) ⊆ p2. From Theorem 4.11 we get a minimal model R → R[U, T ]→ S


where U = U1, together with a minimal model B → B[X,T ]eσ−→ S where (B, q, k)

is a regular local ring and Ker(σ0) ⊆ q2, linked for each n ≥ 2 by isomorphisms

R[U, T ]

(m, U, T<n)R[U, T ]∼=

A[X,U, T ]

(p, X, U, T<n)A[X,U, T ]∼=

B[X,T ]

(q, X, T<n)B[X,T ].

We also have Hi(B[X,T ]/qB[X,T ]) = TorBi (S, k) by definition, and TorBi (S, k) = 0for i > dimB− depthS by the Auslander-Buchsbaum Equality. Thus, Proposition5.2 yields wcat(ϕ) = wcat(R[U, T ]) ≤ dimB − depthS. It remains to note thatdimB = edimS because Ker(σ0) is contained in q2. �

For completeness, we deduce [10, (3.8)] from Proposition 5.2.

5.7. Theorem. If ϕ : (R,m, k)→ S is a local homomorphism, then

wcat(ϕ) ≤ edim(S/mS) + fdR S .

Proof. There is nothing to prove unless f = fdR S is finite. Let R → R′[U ] → Sbe a minimal model of ϕ with edimR′/mR′ = edimS/mS; call this number e. ByProposition 5.2, it suffices to show Hi(R

′[U ]/m′R′[U ]) = 0 for i > e+ f .

Since R′[U ] is a flat resolution of S over R, we have Hi(F ) = TorRi (k, S) = 0 fori > f and F = R′[U ]/mR′[U ]. Now note that F is a complex of free modules overthe regular local ring R = R′/mR′, and that dimR = e by the minimality of theCohen factorization. For i = 0, . . . , e− 1 form exact sequences of complexes

0 −→ F/(x1, . . . , xi)xi+1−−−→ F/(x1, . . . , xi) −→ F/(x1, . . . , xi+1) −→ 0

where x1, . . . , xe is a regular system of parameters of R. From their homologyexact sequences, one sees by induction on i that the homology of the complexF/(x1, . . . , xe) = R′[U ]/m′R′[U ] vanishes in degrees greater than e+ f . �

6. Andre-Quillen homology of local homomorphisms

In this section we prove local versions of our results on Andre-Quillen homology.When using the general properties of the theory we take Andre’s monograph [2] asstandard reference. In addition, we heavily draw on some results from [10], verbatimor in variants. We recall them below.

6.1. For each local homomorphism ψ : Q→ (R,m, k), by [10, (4.3)] one has

rankk Dn(R|Q; k) = εn+1(ψ) for

{2 ≤ n <∞ if chark = 0 ;

2 ≤ n ≤ 2p− 1 if chark = p > 0 .

6.2. For each local homomorphism ψ : Q→ (R,m, k) the following are equivalent.

(i) Dn(R|Q; k) = 0 for all n� 0, and fdQR <∞ .(ii) Dn(R|Q; k) = 0 for all n ≥ 2 .(iii) D2(R|Q; k) = 0 .(iv) Dn(R|Q; k) = 0 for some n ≥ 2 such that bn2 c! 6= 0 ∈ k, and fdQR <∞ .(v) ψ is complete intersection at m .

Indeed, (ii), (iii), and (v) are equivalent by [10, (1.8)]. If ψ is c.i. at m, then fdQRis finite, cf. [11, (3.2)], so (ii) and (v) imply (i) and (iv). Conversely, (i) implies (v)by [10, (4.4)], while (iv) implies (v) by [10, (3.4)] via the equalities in 6.1.


6.3. Let ξ : (R,m, k)→ (R∗,m∗, k∗) be a flat local homomorphism such that m∗ =mR∗. If in a commutative square of local homomorphisms

P

##HHHHHHHHH

Qψ //

ι

;;wwwwwwwwwR

ξ // R∗

the upper path is a regular factorization of the composition ξ◦ψ, then the canonicalmaps Dn(ξ |ι; k∗) : Dn(R|Q; k∗)→ Dn(R

∗ |P ; k∗) are bijective for all n ≥ 2.Indeed, the argument for [10, (1.7)] carries over with only notational changes.

We now present our main local result, describing c.i. homomorphisms in termsparallel to those in 6.2, but without the hypothesis of finite flat dimension.

6.4. Theorem. If ψ : Q → (R,m, k) is a local homomorphism, then the followingconditions are equivalent.

(i) Dn(R|Q; k) = 0 for all n� 0, and ψ is almost small.(ii) Dn(R|Q; k) = 0 for all n� 0 and wcat(ψ) <∞.(iii) Dn(R|Q; k) = 0 for some n ≥ 2 with bn2 c! 6= 0 ∈ k, and ψ is almost small .(iv) Dn(R|Q; k) = 0 for some n ≥ 2 with bn2 c! 6= 0 ∈ k, and wcat(ψ) <∞ .(v) ψ is complete intersection at m .

Proof. If ψ is c.i. at m, then Dn(R|Q; k) = 0 for all n ≥ 2 by 6.2, and ψ is almostsmall by Corollary 4.9, so (v) implies (i) and (iii). If ψ is almost small, then it hasfinite weak category by Theorem 5.6, so (i) implies (ii) and (iii) implies (iv). If (ii)

holds, then the proof of [10, (4.4)] shows that lim supnn√εn(ψ) ≤ 1, so (v) holds

by Corollary 5.5. In view of 6.1, the same corollary shows that (iv) implies (v). �

To continue, we recall some general computations of Andre-Quillen homology.

6.5. Let ϕ : R→ S be a homomorphism of commutative rings, and N an S-module.Andre [1, (16.1)] constructs a universal coefficients spectral sequence

2 Ep,q = TorSp (Dq(S |R;S), N) =⇒ Dp+q(S |R;N) .

Thus, if for some m ∈ Z the S-module Dn(S |R;S) is flat for all n ≤ m, then

(10) Dn(S |R;N) ∼= Dn(S |R;S)⊗S N for all n ≤ m+ 1 .

Set a = Ker(ϕ). There are isomorphisms of S-modules

D1(S |R;N) ∼= a/a2 ⊗R N ∼= TorR1 (S,N) ;(11)

D2(S |R;S) ∼=TorR2 (S, S)

TorR1 (S, S) · TorR1 (S, S),(12)

where the second one is elementary, and the other two come from [2, (6.1), (15.8)].

We give “concrete” descriptions of split c.i. local homomorphisms.

6.6. Let ϕ : (R,m, k) → (S, n, k) be a local homomorphism, set Ker(ϕ) = a, andlet asterisks ∗ denote a-adic completion. If ψ : S → R is a section of ϕ, then thereexist a set x of formal indeterminates over S and a commutative diagram

R∗

ϕ∗

%%KKKKKKKKKKK

Sι //

ψ∗99sssssssssss

S[[x]]/(f)

∼=

OO

π // S


of homomorphisms of rings, where the isomorphism is induced by a surjective mapρ : S[[x]] → R∗ with Ker(ρ) ⊆ n(x) + (x)2, f is a minimal set of generators ofKer(ρ), π is the surjection with Ker(π) = (x), and ι is the natural injection.

Indeed, as S is discrete in the a-adic topology, completion yields homomorphismsof rings ψ∗ : S → R∗ and ϕ∗ : R∗ → S whose composition is the identity mapof S. Pick a set a = {a1, . . . , ae} ⊂ R∗ minimally generating Kerϕ∗, let x ={x1, . . . , xe} be a set of formal indeterminates, and let ρ : S[[x]]→ R∗ be the uniquehomomorphism of S-algebras mapping xi to ai for each i. It is necessarily surjective,and in view of the choice of a we have Ker(ρ) ⊆ (n + (x))2 ∩ (x) = n(x) + (x)2.

6.7. Proposition. In the notation of 6.6 the following hold.

(a) f is a regular sequence if and only if ψ is c.i. at m.(b) f is a regular sequence contained in (x)2 if and only if the S-modules

D1(S |R;S) and D2(S |R;S) are free, and D3(S |R;S) = 0.(c) f = ∅ if and only if the S-module D1(S |R;S) is free, D2(S |R;S) = 0, and

D3(S |R;S) = 0.

Proof. Set P = S[[x]]. The ring R∗ is local with maximal ideal mR∗ and residuefield R∗/mR∗ ∼= k, and R∗ is flat over R. Thus, [2, (4.54)] yields Dn(S |R;S) ∼=Dn(S |R∗;S) for all n ∈ Z, so for the rest of the proof we may assume R = R∗.

As D2(R|S; k) ∼= D2(R|P ; k) by 6.3, (a) follows from 6.2.Since x is P -regular, Dn(S |P ;−) = 0 for n ≥ 2, so the Jacobi-Zariski exact

sequence [2, (5.1)] generated by P → R→ S yields isomorphisms of functors

(13) Dn(S |R;−) ∼= Dn−1(R|P ;−) for n ≥ 3

and, in view of the isomorphism (11), also an exact sequence

(14) 0 −→ D2(S |R;S) −→ ((f )/(f)2)⊗R Sδ−→ (x)/(x)2 −→ D1(S |R;S) −→ 0

of S-modules. Note that the S-module (x)/(x)2 is free, and

(15) Im(δ) = ((f ) + (x)2)/(x)2 ⊆ (n(x) + (x)2)/(x)2 = n((x)/(x)2

).

If f is a regular sequence, then D2(R|P ;S) = 0 and (f )/(f)2 is free over R, sothe “only if” parts of (b) and (c) are now clear. To obtain converses, we assumethat Dn(S |R;S) is free for n ≤ 2, and D3(S |R;S) = 0. From (13) and (10) we get

D2(R|P ; k) ∼= D3(S |R; k) ∼= D3(S |R;S)⊗S k = 0 ,

from where we conclude that f is a regular sequence, cf. 6.2. Due to (14) and(15), the freeness of D1(S |R;S) implies δ = 0, that is, f ⊆ (x)2 and D2(S |R;S) ∼=(f)/(f )2. It is now clear that the “if” parts of (b) and (c) hold as well. �

The next theorem is a local version of Theorem II. The proof draws on resultsobtained above and on earlier results from [13]. One of them is for a homomorphismϕ : (R,m, k)→ (S, n, k) of local rings that is large in the sense of Levin [23], meaning

that the map Torϕ•(ϕ, k) : TorR

•(k, k)→ TorS

•(k, k) is surjective.

6.8. Theorem. For an algebra retract Sψ−→ (R,m, k)

ϕ−→ S of local rings the fol-

lowing conditions are equivalent.

(i) The S-algebra TorR•(S, S) is finitely generated.

(ii) For every S-algebra T there is an isomorphism of graded T -algebras

TorR• (S, T ) ∼=(∧

SD1 ⊗S SymS D2

)⊗S T


where D1 and D2 are free S-modules concentrated in degrees 1 and 2, re-spectively. Moreover, if char(k) > 0, then D2 = 0.

(iii) The S-modules D1(S |R;S) and D2(S |R;S) are free, and D3(S |R;S) = 0.Moreover, if char(k) > 0, then D2(S |R;S) = 0.

(iv) The sequence f in the commutative diagram constructed in 6.6 is containedin (x)2. Moreover, if char(k) > 0, then f = ∅.

Proof. Suppose that char(k) > 0. Conditions (iii) and (iv) are then equivalent byProposition 6.7.c. Functoriality in the ring argument shows that ϕ is large. Forevery large homomorphism, conditions (i), (ii) and (iv) are equivalent by [13, (3.1)].

Now consider the case when char(k) = 0. Conditions(iii) and (iv) are equivalentby Proposition 6.7.b. The equivalence of (i), (ii), and (iv) is established in [13,(4.1)] under an additional hypothesis, R has finite flat dimension over S, which isused only once, to conclude that Dn(R

∗ |S[[x]]; k) = 0 for all n� 0 implies that f

is regular (cf. [13, p. 163]). By Theorem 6.4, the same conclusion holds when themap ρ : S[[x]] → R∗ is almost small. To see that it is, note that the composition

S[[x]]ρ−→ R∗ ψ∗

−−→ S is complete intersection at n, and apply Corollary 4.9. �

The proof above raises the question whether the first three conditions are equiv-alent for large homomorphisms in characteristic 0. Here is what we know.

6.9. Let ϕ : R → (S, n, k) be a large homomorphism, where char(k) = 0. If R →R[U ]→ S is a minimal model of ϕ, then results of Quillen, [25, (9.5), (10.1)], imply

sup{n ∈ N | Dn(S |R; k) 6= 0} = sup{n ∈ N | Un 6= ∅} .

Thus, [13, (3.2)] shows that if TorR• (S, S) is finitely generated, then AQ-dimS R isfinite, and [21, (3.1)] proves that in this case AQ-dimS R is equal to 1 or is even.

7. Andre-Quillen dimension

In this section we prove the theorems stated in the Introduction. The local casewas essentially settled in Section 6, but reduction to that case needs some attention,as weak category and almost small homomorphism are intrinsically local notions.

We start by recording a slight extension of a result of Andre [2]. For eachn ∈ SpecS, we let k(n) denote the residue field Sn/nSn.

7.1. If Q→ R→ S are homomorphisms of noetherian rings and j is a non-negativeinteger, then the following conditions are equivalent.

(i) Dn(R|Q;−) = 0 on the category of S-modules for all n ≥ j.(ii) Dn(Rm |Qn∩Q; k(m)) = 0 for all n ≥ j, all n ∈ SpecS, and m = n ∩R.

Indeed, [2, (4.58), (5.27)] produce for each n canonical isomorphisms

Dn(R|Q; k(n)) ∼= Dn(Rm |Q; k(m))⊗k(m) k(n) ∼= Dn(Rm |Qn∩Q; k(m))⊗k(m) k(n)

which show that (i) implies (ii). If R → S is the identity map, then (ii) implies(i) by [2, Suppl. Prop. 29]; the proof of that proposition applies verbatim to everyhomomorphism R→ S in which the ring S is noetherian.

We recall from [10] the definition of l.c.i. homomorphism of noetherian rings.

7.2. Definition. A homomorphism of noetherian rings ϕ : R → S is called locallycomplete intersection (or l.c.i.) if for every n ∈ SpecS the induced local homomor-phism ϕn : Rn∩R → Sn is complete intersection at nSn in the sense of 2.7.


We return to the discussion, started in the Introduction, of criteria for l.c.i.homomorphisms in terms of Andre-Quillen homology. Recall that ϕ : R → S issaid to be locally of finite flat dimension if fdR(Sn) <∞ for every n ∈ SpecS.

7.3. If ϕ : R → S is a homomorphism of noetherian rings, then the following con-ditions are equivalent.

(i) AQ-dimR S <∞, and ϕ is locally of finite flat dimension.(ii) AQ-dimR S ≤ 1 .(iii) D2(S |R;−) = 0 .(iv) Dn(S |R;−) = 0 for some n ≥ 2 with bn2 c! invertible in S, and ϕ is locally

of finite flat dimension.(v) ϕ is locally complete intersection .

Indeed, (v) is equivalent to (ii), (iii) by [10, (1.2)], and to (i), (iv) by [10, (1.5)].

A major difficulty in dealing with Quillen’s Conjecture is that R-algebras S withAQ-dimR S ≤ 2 have not been described in structural terms. All known algebrassatisfying this condition are constructed by factoring some l.c.i. homomorphism.

7.4. Example. Let Q → S be an l.c.i. homomorphism from a noetherian ring Q(say, a surjection with kernel generated by a regular sequence g), through somehomomorphism Q → R that is c.i. at all primes of R contracted from S (say,a surjection with kernel generated by a regular sequence in (g)). The desiredvanishing property follows from 7.3, via the Jacobi-Zariski exact sequence [2, (5.1)].

The construction described above is rather rigid. This is demonstrated by thenext result, which could be compared with another factorization theorem for localhomomorphisms from [10]: If ϕ ◦ ψ is l.c.i. and ϕ is locally of finite flat dimension,then ϕ is l.c.i. and ψ is complete intersection at n ∩R for each n ∈ SpecS.

7.5. Theorem. If ψ : Q → R and ϕ : R → S are homomorphisms of noetherianrings such that ϕ ◦ ψ is l.c.i., then the following conditions are equivalent.

(i) AQ-dimR S <∞.(ii) AQ-dimR S ≤ 2 .(iii) D3(S |R;−) = 0 .(iv) Dn(S |R;−) = 0 for some n ≥ 3 such that bn−1

2 c! is invertible in S .(v) ψ is complete intersection at n ∩R for each n ∈ SpecS.

Proof. The Jacobi-Zariski exact sequence of Andre-Quillen homology, cf. [2, (5.1)],yields an exact sequence of functors on the category of S-modules

· · · −→ Dn+1(S |Q;−) −→ Dn+1(S |R;−) −→ Dn(R|Q;−) −→ Dn(S |Q;−) −→ · · ·

Since ϕ ◦ ψ is l.c.i., we have Dn(S |Q;−) = 0 for n ≥ 2 by 7.3, so for n ≥ 2 we getisomorphisms Dn+1(S |R;−) ∼= Dn(R|Q;−) of functors of S-modules. Thus, eachone of conditions (i), (ii), and (iii) is equivalent to its primed version below:

(i′) Dn(R|Q;−) = 0 on the category of S-modules for all n� 0.(ii′) Dn(R|Q;−) = 0 on the category of S-modules for all n ≥ 2.(iii′) D2(R|Q;−) = 0 on the category of S-modules.(iv′) Dn(R|Q;−) = 0 on the category of S-modules for some n ≥ 2 such that

bn2 c! is invertible in S.

The equivalence of (ii′) and (v) results from 7.1 and 6.2. For each n ∈ SpecS,the homomorphism ψn∩R : Qn∩Q → Rn∩R is almost small by Corollary 4.9. Theequivalence of (i′), (iii′), (iv′), and (v) now comes from 7.1 and Theorem 6.4. �


The last result implies part of [10, (1.5)]: Quillen’s Conjecture holds when thering S is l.c.i., because the theorem then applies with Q = Z. More to the point, itreduces the proof of Theorem I from the Introduction to a mere formality.

Proof of Theorem I. Let ψ : S → R be any section of ϕ.The map ϕ ◦ ψ = idS is obviously l.c.i., so Theorem 7.5 and Proposition 6.7.a

show that the conditions of Theorem I are equivalent. �

It remains to deduce Theorem II from its local version established in Section 6.

Proof of Theorem II. The implication (ii) =⇒ (i) is clear. By (11) and (12), the S-modules D1(R|S;S) and D2(R|S;S) are finitely generated. Thus the implications(i) =⇒ (iii) ⇐⇒ (iv) follow from Theorem 6.8 via the isomorphisms

TorR• (S, S)n∼= TorRn∩R

• (Sn, Sn) ;

D•(S |R;S)n∼= D•(Sn |Rn∩R;Sn) ,

respectively of graded Sn-algebras and graded Sn-modules, cf. [2, (4.59), (5.27)].It remains to prove (iii) =⇒ (ii). Let D• denote the graded S-module with

Dn = Dn(S |R;S) for n = 1, 2, and Dn = 0 otherwise. The isomorphisms (11)

and (12) define a surjection τ : TorR• (S, S)→ D• of graded S-modules. Since D• is

projective, we may choose an S-linear map θ : D• → TorR• (S, S) with τ ◦ θ = idD• .It extends to a homomorphism of graded S-algebras

ϑ :∧SD1 ⊗S SymS D2 −→ TorR

•(S, S) .

Theorem 6.8 and the isomorphisms above show that ϑn is bijective for every n ∈MaxS, so ϑ is an isomorphism. It induces an isomorphism

ϑ⊗S T : (∧SD1 ⊗S SymS D2)⊗S T

∼=−−→ TorR• (S, S)⊗S T

of graded T -algebras. As each S-module TorRn (S, S) is projective, the UniversalCoefficients Theorem shows that for every S-algebra T the homomorphism

TorR•(S, S)⊗S T

∼=−−→ TorR•(S, T )

of graded T -algebras given by the Kunneth map is bijective. Composing the iso-morphisms above we obtain the desired isomorphism of graded T -algebras. �

We finish the paper by revisiting the announcement in [14].

7.6. Remark. In [14, (2.6)], which is an avatar of Theorem 7.5, instead of condition(7.5.i) one finds the seemingly weaker condition: AQ-dimR Sn < ∞ for all n ∈SpecS. We show that this condition is in fact equivalent to those in Theorem 7.5.

Indeed, under the hypotheses of Theorem 7.5 (which coincide with those [14,(2.6)]) the composition of the maps ψ : Q → R and R → Sn is an l.c.i. homomor-phism. Fixing a prime ideal n of S, from Theorem 7.5 we see that AQ-dimR Sn isfinite if and only if ψ is c.i. at q ∩ R for all q ∈ SpecS with q ⊆ n. Letting now n

range over SpecS, we conclude that AQ-dimR Sn is finite for all n ∈ SpecS if andonly if ψ is c.i. at q ∩R for all q ∈ SpecS. This is condition (7.5.v).

We take this opportunity to make two minor corrections to [14]:

– in [14, (2.6.i)] ‘at each p ∈ SpecR with p ⊇ Ker(ϕ)’ should read ‘at n ∩ R foreach n ∈ SpecS’;

– in [14, line above (4.1)] ‘in [13, (4.1.iii)]’ should read ‘in [13, Theorem II]’.

We thank the editors of the present article for pointing these errors to us.


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Department of Mathematics, Purdue University, West Lafayette, IN 47907, USAE-mail address: [email protected]

Current address: Department of Mathematics, University of Nebraska, Lincoln, NE 68588,USA

Pure Mathematics, Hicks Building, University of Sheffield, Sheffield S3 7RH, UKE-mail address: [email protected]

Current address: Department of Mathematics, University of Missouri, Columbia, MO 65201,USA

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